Jmat.Real.erf

Average Error: 13.9 → 13.9

Time: 2.3m

Precision: 64

Math

\[1 - \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(0.25482959199999999 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-0.284496735999999972 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(1.42141374100000006 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-1.45315202700000001 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot 1.0614054289999999\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]

↓

\[\left(\sqrt[3]{1 - \left(\frac{1}{1 \cdot 1 - \left(0.32759110000000002 \cdot \left|x\right|\right) \cdot \left(0.32759110000000002 \cdot \left|x\right|\right)} \cdot \left(\left(1 - 0.32759110000000002 \cdot \left|x\right|\right) \cdot \left(\left(0.25482959199999999 + -0.284496735999999972 \cdot \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}\right) + \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(1.42141374100000006 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-1.45315202700000001 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot 1.0614054289999999\right)\right)\right) \cdot \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}} \cdot \sqrt[3]{1 - \left(\frac{1}{1 \cdot 1 - \left(0.32759110000000002 \cdot \left|x\right|\right) \cdot \left(0.32759110000000002 \cdot \left|x\right|\right)} \cdot \left(\left(1 - 0.32759110000000002 \cdot \left|x\right|\right) \cdot \left(\left(0.25482959199999999 + -0.284496735999999972 \cdot \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}\right) + \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(1.42141374100000006 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-1.45315202700000001 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot 1.0614054289999999\right)\right)\right) \cdot \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}}\right) \cdot \sqrt[3]{1 - \left(\frac{1}{1 \cdot 1 - \left(0.32759110000000002 \cdot \left|x\right|\right) \cdot \left(0.32759110000000002 \cdot \left|x\right|\right)} \cdot \left(\left(1 - 0.32759110000000002 \cdot \left|x\right|\right) \cdot \left(\left(0.25482959199999999 + -0.284496735999999972 \cdot \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}\right) + \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(1.42141374100000006 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-1.45315202700000001 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot 1.0614054289999999\right)\right)\right) \cdot \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}}\]

C

double f(double x) {
        double r706621 = 1.0;
        double r706622 = 0.3275911;
        double r706623 = x;
        double r706624 = fabs(r706623);
        double r706625 = r706622 * r706624;
        double r706626 = r706621 + r706625;
        double r706627 = r706621 / r706626;
        double r706628 = 0.254829592;
        double r706629 = -0.284496736;
        double r706630 = 1.421413741;
        double r706631 = -1.453152027;
        double r706632 = 1.061405429;
        double r706633 = r706627 * r706632;
        double r706634 = r706631 + r706633;
        double r706635 = r706627 * r706634;
        double r706636 = r706630 + r706635;
        double r706637 = r706627 * r706636;
        double r706638 = r706629 + r706637;
        double r706639 = r706627 * r706638;
        double r706640 = r706628 + r706639;
        double r706641 = r706627 * r706640;
        double r706642 = r706624 * r706624;
        double r706643 = -r706642;
        double r706644 = exp(r706643);
        double r706645 = r706641 * r706644;
        double r706646 = r706621 - r706645;
        return r706646;
}

↓

double f(double x) {
        double r706647 = 1.0;
        double r706648 = r706647 * r706647;
        double r706649 = 0.3275911;
        double r706650 = x;
        double r706651 = fabs(r706650);
        double r706652 = r706649 * r706651;
        double r706653 = r706652 * r706652;
        double r706654 = r706648 - r706653;
        double r706655 = r706647 / r706654;
        double r706656 = r706647 - r706652;
        double r706657 = 0.254829592;
        double r706658 = -0.284496736;
        double r706659 = r706647 + r706652;
        double r706660 = r706647 / r706659;
        double r706661 = r706658 * r706660;
        double r706662 = r706657 + r706661;
        double r706663 = 1.421413741;
        double r706664 = -1.453152027;
        double r706665 = 1.061405429;
        double r706666 = r706660 * r706665;
        double r706667 = r706664 + r706666;
        double r706668 = r706660 * r706667;
        double r706669 = r706663 + r706668;
        double r706670 = r706660 * r706669;
        double r706671 = r706670 * r706660;
        double r706672 = r706662 + r706671;
        double r706673 = r706656 * r706672;
        double r706674 = r706655 * r706673;
        double r706675 = r706651 * r706651;
        double r706676 = -r706675;
        double r706677 = exp(r706676);
        double r706678 = r706674 * r706677;
        double r706679 = r706647 - r706678;
        double r706680 = cbrt(r706679);
        double r706681 = r706680 * r706680;
        double r706682 = r706681 * r706680;
        return r706682;
}

Error

Bits error versus x

Derivation

1. Initial program 13.9

   \[1 - \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(0.25482959199999999 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-0.284496735999999972 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(1.42141374100000006 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-1.45315202700000001 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot 1.0614054289999999\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]
2. Using strategy rm

3. Applied distribute-rgt-in 13.9

   \[\leadsto 1 - \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(0.25482959199999999 + \color{blue}{\left(-0.284496735999999972 \cdot \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} + \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(1.42141374100000006 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-1.45315202700000001 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot 1.0614054289999999\right)\right)\right) \cdot \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}\right)}\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]

4. Applied associate-+r+ 13.9

   \[\leadsto 1 - \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \color{blue}{\left(\left(0.25482959199999999 + -0.284496735999999972 \cdot \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}\right) + \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(1.42141374100000006 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-1.45315202700000001 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot 1.0614054289999999\right)\right)\right) \cdot \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}\right)}\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]
5. Using strategy rm

6. Applied flip-+ 13.9

   \[\leadsto 1 - \left(\frac{1}{\color{blue}{\frac{1 \cdot 1 - \left(0.32759110000000002 \cdot \left|x\right|\right) \cdot \left(0.32759110000000002 \cdot \left|x\right|\right)}{1 - 0.32759110000000002 \cdot \left|x\right|}}} \cdot \left(\left(0.25482959199999999 + -0.284496735999999972 \cdot \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}\right) + \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(1.42141374100000006 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-1.45315202700000001 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot 1.0614054289999999\right)\right)\right) \cdot \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]

7. Applied associate-/r/ 13.9

   \[\leadsto 1 - \left(\color{blue}{\left(\frac{1}{1 \cdot 1 - \left(0.32759110000000002 \cdot \left|x\right|\right) \cdot \left(0.32759110000000002 \cdot \left|x\right|\right)} \cdot \left(1 - 0.32759110000000002 \cdot \left|x\right|\right)\right)} \cdot \left(\left(0.25482959199999999 + -0.284496735999999972 \cdot \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}\right) + \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(1.42141374100000006 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-1.45315202700000001 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot 1.0614054289999999\right)\right)\right) \cdot \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]

8. Applied associate-*l* 13.9

   \[\leadsto 1 - \color{blue}{\left(\frac{1}{1 \cdot 1 - \left(0.32759110000000002 \cdot \left|x\right|\right) \cdot \left(0.32759110000000002 \cdot \left|x\right|\right)} \cdot \left(\left(1 - 0.32759110000000002 \cdot \left|x\right|\right) \cdot \left(\left(0.25482959199999999 + -0.284496735999999972 \cdot \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}\right) + \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(1.42141374100000006 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-1.45315202700000001 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot 1.0614054289999999\right)\right)\right) \cdot \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}\right)\right)\right)} \cdot e^{-\left|x\right| \cdot \left|x\right|}\]
9. Using strategy rm

10. Applied add-cube-cbrt 13.9

    \[\leadsto \color{blue}{\left(\sqrt[3]{1 - \left(\frac{1}{1 \cdot 1 - \left(0.32759110000000002 \cdot \left|x\right|\right) \cdot \left(0.32759110000000002 \cdot \left|x\right|\right)} \cdot \left(\left(1 - 0.32759110000000002 \cdot \left|x\right|\right) \cdot \left(\left(0.25482959199999999 + -0.284496735999999972 \cdot \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}\right) + \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(1.42141374100000006 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-1.45315202700000001 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot 1.0614054289999999\right)\right)\right) \cdot \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}} \cdot \sqrt[3]{1 - \left(\frac{1}{1 \cdot 1 - \left(0.32759110000000002 \cdot \left|x\right|\right) \cdot \left(0.32759110000000002 \cdot \left|x\right|\right)} \cdot \left(\left(1 - 0.32759110000000002 \cdot \left|x\right|\right) \cdot \left(\left(0.25482959199999999 + -0.284496735999999972 \cdot \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}\right) + \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(1.42141374100000006 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-1.45315202700000001 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot 1.0614054289999999\right)\right)\right) \cdot \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}}\right) \cdot \sqrt[3]{1 - \left(\frac{1}{1 \cdot 1 - \left(0.32759110000000002 \cdot \left|x\right|\right) \cdot \left(0.32759110000000002 \cdot \left|x\right|\right)} \cdot \left(\left(1 - 0.32759110000000002 \cdot \left|x\right|\right) \cdot \left(\left(0.25482959199999999 + -0.284496735999999972 \cdot \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}\right) + \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(1.42141374100000006 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-1.45315202700000001 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot 1.0614054289999999\right)\right)\right) \cdot \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}}}\]

11. Final simplification 13.9

    \[\leadsto \left(\sqrt[3]{1 - \left(\frac{1}{1 \cdot 1 - \left(0.32759110000000002 \cdot \left|x\right|\right) \cdot \left(0.32759110000000002 \cdot \left|x\right|\right)} \cdot \left(\left(1 - 0.32759110000000002 \cdot \left|x\right|\right) \cdot \left(\left(0.25482959199999999 + -0.284496735999999972 \cdot \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}\right) + \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(1.42141374100000006 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-1.45315202700000001 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot 1.0614054289999999\right)\right)\right) \cdot \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}} \cdot \sqrt[3]{1 - \left(\frac{1}{1 \cdot 1 - \left(0.32759110000000002 \cdot \left|x\right|\right) \cdot \left(0.32759110000000002 \cdot \left|x\right|\right)} \cdot \left(\left(1 - 0.32759110000000002 \cdot \left|x\right|\right) \cdot \left(\left(0.25482959199999999 + -0.284496735999999972 \cdot \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}\right) + \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(1.42141374100000006 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-1.45315202700000001 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot 1.0614054289999999\right)\right)\right) \cdot \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}}\right) \cdot \sqrt[3]{1 - \left(\frac{1}{1 \cdot 1 - \left(0.32759110000000002 \cdot \left|x\right|\right) \cdot \left(0.32759110000000002 \cdot \left|x\right|\right)} \cdot \left(\left(1 - 0.32759110000000002 \cdot \left|x\right|\right) \cdot \left(\left(0.25482959199999999 + -0.284496735999999972 \cdot \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}\right) + \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(1.42141374100000006 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-1.45315202700000001 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot 1.0614054289999999\right)\right)\right) \cdot \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}}\]

Reproduce

herbie shell --seed 2020003 
(FPCore (x)
  :name "Jmat.Real.erf"
  :precision binary64
  (- 1 (* (* (/ 1 (+ 1 (* 0.3275911 (fabs x)))) (+ 0.254829592 (* (/ 1 (+ 1 (* 0.3275911 (fabs x)))) (+ -0.284496736 (* (/ 1 (+ 1 (* 0.3275911 (fabs x)))) (+ 1.421413741 (* (/ 1 (+ 1 (* 0.3275911 (fabs x)))) (+ -1.453152027 (* (/ 1 (+ 1 (* 0.3275911 (fabs x)))) 1.061405429))))))))) (exp (- (* (fabs x) (fabs x)))))))